Paul Seymour is a mathematician working in graph theory, the study of network interconnections. He has been awarded several prizes, including the Ostrowski Prize, and the D. R. Fulkerson Prize (four times). Here are two of his most important results.
With Neil Robertson, Seymour proved the 'graph minors structure theorem', which says that for any graph, all graphs that do not contain it as a 'minor' can more-or-less be drawn on a surface of bounded genus. This has had many applications: for instance, it implied that in any infinite set of graphs, one of them contains another as a minor.
With Maria Chudnovsky, Neil Robertson and Robin Thomas, Seymour proved the 'strong perfect graph theorem'. 'Colouring' a graph means giving each vertex a colour different from the colours of its neighbours, and the game is to use the fewest possible number of colours. If the graph contains, say, ten vertices all joined to one another, then at least ten colours will be needed, and perhaps more: but sometimes the maximum number of vertices all joined to one another is the right answer. They characterized when this is true, which was a famous, long-standing and much-studied question.